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2.3 – Formal Proofs
1.
State and prove:
SAS
2. ASA
3. AAS
4. SSS
(NOTE: The congruence results above are independent of the parallel postulate)
2.
State and prove the Isosceles Triangle Theorem (ITT) for planes. If we assumed
Euclid’s development of axioms and theorems to prove this result, we would be using
circular reasoning – Explain why.
3.
State the converse of the ITT.
4.
Use the ITT to show that the bisector of the top angle of an isosceles triangle is
also the perpendicular bisector of the base of that triangle.
5.
Show that for two right triangles, if the hypotenuse and leg of one triangle are
congruent to the hypotenuse and leg of the other, then the two triangles are congruent
[Try a proof by contradiction].
6.
Give counterexamples to show that generally SAS is not true on spheres.
Comment on ASA, AAS, and SSS on spheres.
7.
Define the terms: ‘quadrilateral’, ‘sides of a quadrilateral’, diagonals of a
quadrilateral’, ‘parallelogram’, ‘height of a parallelogram’, ‘two equivalent geometric
figures’, ‘rectangle’, ‘base and height of a rectangle’, ‘area of a rectangle’
8.
When are two figures said to have the same area?
9.
Prove: The area of a right triangle is one-half the product of the lengths of its legs.
10.
Prove: The area of any triangle is one-half the product of a base of the triangle
with the height of the triangle.
11.
Define ‘cevian’ of a triangle.
12.
State and prove Ceva’s Theorem and its converse.
13.
Define the terms: ‘median’ of a triangle.
14.
Prove: The medians of a triangle intersect at a common point called the centroid
of the triangle.
15.
Definition: Two triangles are similar, if and only if there is some way to match
the vertices of one triangle to those of the other such that the corresponding sides are in
the same ratio and corresponding angles are congruent. One triangle is a scaled- up
version of the other. The scaling factor is the constant of proportionality between the
corresponding sides.
16.
Theorem 1: Suppose a line is parallel to one side of a triangle and intersects the
other two sides in different points. Then, this line divides the intersected sides into
proportional segments
A
D
E
l
B
C
( Given l is parallel to BC, we claim:
BD CE

)
AD AE
AB AC

AD AE
18.
State and prove the converse of Theorem 1
19.
Theorem 2: (AAA similarity condition) If in two triangles there is a
correspondence in which the three angles of one triangle are congruent to the three angles
of the other triangle, then the triangles are similar.
20.
Theorem 2: (SAS similarity condition) If in two triangles there is a
correspondence in which two sides of triangle are proportional to two sides of the other
triangles and the included angles are congruent, then the triangles are similar.
21.
Theorem 3: (SSS similarity condition). State and prove the SSS similarity
condition.
22.
Exercise 3: State and prove Menelaus’ Theorem (Exercise 2.5.7 in text)
23.
Define the following terms:
17.
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Prove
Chord of a circle
Diameter of a circle
Open arc of a circle
Semi-circle
Closed semi-circle
Minor arc
Major arc
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Central angle of a circle
Inscribed angle of a circle
Tangent to a circle
Point of tangency
Mutually tangent circles
Externally tangent circles
Internally tangent circles
Prove:
a) Given three distinct points, not all on the same line, there is a unique circle through
these points
b) The measure of an angle inscribed in a circle is half that of its intercepted central
angle
c) If two angles are inscribed in a circle such that they share the same arc, then the
angles are congruent
d) An inscribed angle in a semi-circle is always a right angle.
e) If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.
f) If an inscribed angle in a circle is a right angle, then the endpoints of the arc are on a
diameter.
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g) Given a circle c with center O and radius OT , a line l is tangent to c at T if and only
____
if l is perpendicular to OT at T.
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h) Given a circle c with center O and radius OA and given a point P outside of the
circle, there are exactly two tangent lines to the circle passing through P